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Properties

Label 4-855e2-1.1-c1e2-0-2
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 46.610746.6107
Root an. cond. 2.612892.61289
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 2·5-s − 2·7-s − 6·11-s − 6·13-s + 5·16-s + 8·17-s − 2·19-s − 6·20-s − 8·23-s + 3·25-s − 6·28-s − 2·29-s + 12·31-s + 4·35-s + 2·37-s + 14·41-s + 6·43-s − 18·44-s − 8·47-s − 4·49-s − 18·52-s − 12·53-s + 12·55-s − 12·59-s − 12·61-s + 3·64-s + ⋯
L(s)  = 1  + 3/2·4-s − 0.894·5-s − 0.755·7-s − 1.80·11-s − 1.66·13-s + 5/4·16-s + 1.94·17-s − 0.458·19-s − 1.34·20-s − 1.66·23-s + 3/5·25-s − 1.13·28-s − 0.371·29-s + 2.15·31-s + 0.676·35-s + 0.328·37-s + 2.18·41-s + 0.914·43-s − 2.71·44-s − 1.16·47-s − 4/7·49-s − 2.49·52-s − 1.64·53-s + 1.61·55-s − 1.56·59-s − 1.53·61-s + 3/8·64-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 46.610746.6107
Root analytic conductor: 2.612892.61289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :1/2,1/2), 1)(4,\ 731025,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2689300061.268930006
L(12)L(\frac12) ≈\approx 1.2689300061.268930006
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good2C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7D4D_{4} 1+2T+8T2+2pT3+p2T4 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+6T+24T2+6pT3+p2T4 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4}
13D4D_{4} 1+6T+28T2+6pT3+p2T4 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23D4D_{4} 1+8T+34T2+8pT3+p2T4 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+2T−4T2+2pT3+p2T4 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4}
31C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
37D4D_{4} 1−2T+68T2−2pT3+p2T4 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−14T+124T2−14pT3+p2T4 1 - 14 T + 124 T^{2} - 14 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−6T+88T2−6pT3+p2T4 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+8T+82T2+8pT3+p2T4 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+12T+114T2+12pT3+p2T4 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+12T+126T2+12pT3+p2T4 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+12T+130T2+12pT3+p2T4 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−8T+38T2−8pT3+p2T4 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+4T+118T2+4pT3+p2T4 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79D4D_{4} 1+8T+62T2+8pT3+p2T4 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
89D4D_{4} 1−18T+196T2−18pT3+p2T4 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−10T+156T2−10pT3+p2T4 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.67271511362287841677187630475, −10.02581513974420537155598790993, −9.579828267433626812097760759297, −9.516934784813309500688880154964, −8.419923197513881581540426457735, −7.80089205681565877940778521648, −7.80062728705847226041248418507, −7.71036418310382896784177414998, −7.11459669919728983154610827023, −6.29703094192450178260044183745, −6.27725230823798878770474023582, −5.71537394554387100462234331872, −5.05133025326034824774734512449, −4.64581092554730096159220447587, −4.02563579301204193992383587525, −3.08520710724224944364398989554, −2.98971781948011649321538550987, −2.50442802574960914334913958952, −1.79827696998905858967763090168, −0.50982924289492545929628281306, 0.50982924289492545929628281306, 1.79827696998905858967763090168, 2.50442802574960914334913958952, 2.98971781948011649321538550987, 3.08520710724224944364398989554, 4.02563579301204193992383587525, 4.64581092554730096159220447587, 5.05133025326034824774734512449, 5.71537394554387100462234331872, 6.27725230823798878770474023582, 6.29703094192450178260044183745, 7.11459669919728983154610827023, 7.71036418310382896784177414998, 7.80062728705847226041248418507, 7.80089205681565877940778521648, 8.419923197513881581540426457735, 9.516934784813309500688880154964, 9.579828267433626812097760759297, 10.02581513974420537155598790993, 10.67271511362287841677187630475

Graph of the ZZ-function along the critical line